##  [Complete Theory](/complete-theory-1) 

 Definition

A first-order theory T in a language L is complete if for every L-sentence φ either T proves φ or T proves ¬φ; equivalently, any two models of T are elementarily equivalent (T decides every sentence of the language).

 

 

 

 

 

 





## Principle

Principle

Completeness expresses that the theory assigns a definite first-order truth-value to every sentence of its language, so syntactic deduction and semantic truth coincide at the level of sentences for that theory.

 

 

 

 

 





## Demonstration

Demonstration

The theory of dense linear orders without endpoints (DLO) is complete: every sentence in the language of linear orders is either provable from DLO or its negation is provable, and all countable dense orders without endpoints are isomorphic to (Q,<), so DLO decides first-order sentences about order-denseness and endpoints.

 

 

 

 

## Misapplication

Misapplication

Confusing completeness with decidability (a complete theory may be undecidable) or with model-completeness (a complete theory need not be model-complete); assuming completeness transfers automatically across expansions of the language without care.

 

 

 

 

 





## Consequence

Consequence

A complete theory yields a fixed first-order picture of its models: types over the empty set are determined, and any two models satisfy exactly the same sentences, facilitating classification and transfer of properties between models.

 

 

 

 

## Reversal

Reversal

An incomplete theory leaves some sentences independent (neither provable nor refutable), allowing multiple non-elementarily-equivalent models and genuine ambiguity in first-order truths about the language.

 

 

 

 

 





## Boundary

Boundary

Completeness is relative to the chosen language and signature: adding symbols can break completeness; completeness does not imply stronger metatheoretic properties like decidability or categoricity without extra hypotheses.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension with model-completeness and decidability: model-complete theories have different embedding behavior than complete ones, and decidability concerns effective determination of provability, a distinct notion from syntactic completeness.

 

 

 

 

 





## Synthesis

Synthesis

A complete theory is a first-order theory that decides every sentence in its language, giving a determinate semantic profile to all its models though not necessarily yielding algorithmic decidability or uniqueness of models in each cardinality.